Transgression and Clifford algebras

@article{Rohr2009,
abstract = {Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, \dots , p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/$$<\!p_1, \dots , p_r\!>)$ is isomorphic to a Clifford algebra $\text\{Cl\}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra $\{\mathcal\{W\}(\mathfrak\{g\})\} = U\mathfrak\{g\} \otimes \text\{Cl\}\mathfrak\{g\}$ for $\mathfrak\{g\}$ a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman ).},
affiliation = {University of Geneva Department of Mathematics 2-4 rue du Lièvre, c.p. 64 1211 Geneva 4 (Suisse)},
author = {Rohr, Rudolf Philippe},
journal = {Annales de l’institut Fourier},
keywords = {Lie algebras; Weil algebras; quantized Weil algebras; Clifford algebras; Transgression; classical and quantized Weil algebras; transgression in spectral sequence},
language = {eng},
number = {4},
pages = {1337-1358},
publisher = {Association des Annales de l’institut Fourier},
title = {Transgression and Clifford algebras},
url = {http://eudml.org/doc/10430},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Rohr, Rudolf Philippe
TI - Transgression and Clifford algebras
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 4
SP - 1337
EP - 1358
AB - Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, \dots , p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/$$<\!p_1, \dots , p_r\!>)$ is isomorphic to a Clifford algebra $\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra ${\mathcal{W}(\mathfrak{g})} = U\mathfrak{g} \otimes \text{Cl}\mathfrak{g}$ for $\mathfrak{g}$ a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman ).
LA - eng
KW - Lie algebras; Weil algebras; quantized Weil algebras; Clifford algebras; Transgression; classical and quantized Weil algebras; transgression in spectral sequence
UR - http://eudml.org/doc/10430
ER -